Weil-\'etale Cohomology over $p$-adic Fields

TL;DR

This paper develops duality theorems for Weil group cohomology over p-adic fields, generalizing Galois cohomology results and introducing Weil-smooth cohomology for varieties, with applications to curves.

Contribution

It establishes new duality theorems for Weil group cohomology and defines Weil-smooth cohomology, extending classical duality results to a broader context.

Findings

Proves a duality theorem for discrete Weil modules, implying Tate-Nakayama Duality.

Defines Weil-smooth cohomology for varieties over local fields.

Establishes a duality theorem for the cohomology of $ ext{G}_m$ on smooth, proper curves.

Abstract

We establish duality results for the cohomology of the Weil group of a $p$-adic field, analogous to, but more general than, results from Galois cohomology. We prove a duality theorem for discrete Weil modules, which implies Tate-Nakayama Duality. We define Weil-smooth cohomology for varieties over local fields, and prove a duality theorem for the cohomology of $\G_m$ on a smooth, proper curve with a rational point. This last theorem is analogous to, and implies, a classical duality theorem for such curves.